Spanning trees and continued fractions

TL;DR

This paper proves that the number of spanning trees in certain graphs grows exponentially with the number of vertices, using a novel connection with continued fractions and number theory.

Contribution

It establishes the exponential growth of spanning trees in simple planar graphs, answering a long-standing question from 1969, and introduces a new approach involving continued fractions.

Findings

Number of spanning trees grows exponentially with vertices

Connection established between spanning trees and continued fractions

Answers Sedlák's 1969 question

Abstract

We prove the exponential growth of the cardinality of the set of numbers of spanning trees in simple (and planar) graphs on $n$ vertices, answering a question of Sedl\'a\v{c}ek from 1969. The proof uses a connection with continued fractions, ``thin orbits,'' and Zaremba's conjecture.